20

Step 1: Convert the penetration distance from mm to m: 18 mm = 0.018 m.
Step 2: Use the work-energy principle. The work done by the axe head in penetrating the tree is equal to the force multiplied by the distance it penetrates:
$$ W = F imes d $$
where F = 80,000 N (force) and d = 0.018 m.
Step 3: Calculate the work done:
$$ W = 80,000 imes 0.018 = 1,440 ext{ J} $$
Step 4: The work done is also equal to the kinetic energy of the axe head just before impact:
$$ W = \frac{1}{2} mv^2 $$
where m = 5 kg, and v is the speed just before impact.
Step 5: Rearranging for v gives:
$$ v = \sqrt{\frac{2W}{m}} $$
Step 6: Substitute W and m into the equation:
$$ v = \sqrt{\frac{2 \times 1440}{5}} = \sqrt{576} = 24 ext{ m/s} $$
Hence, the speed of the axe head just before impact is 24 m/s.